ONE CUBIC DIOPHANTINE INEQUALITY

Abstract

Suppose that G(x) is a form, or homogeneous polynomial, of odd degree d in s variables, with real coefficients. Schmidt [15] has shown that there exists a positive integer s₀(d), which depends only on the degree d, so that if s [ges ] s₀(d), then there is an x ∈ ℤ^s\{0} satisfying the inequality

formula here

In other words, if there are enough variables, in terms of the degree only, then there is a nontrivial solution to (1). Let s₀(d) be the minimum integer with the above property. In the course of proving this important result, Schmidt did not explicitly give upper bounds for s₀(d). His methods do indicate how to do so, although not very efficiently. However, in fact much earlier, Pitman [13] provided explicit bounds in the case when G is a cubic. We consider a general cubic form F(x) with real coefficients, in s variables, and look at the inequality

formula here

Specifically, Pitman showed that if

formula here

then inequality (2) is non-trivially soluble in integers. We present the following improvement of this bound.